Question 1

Q1: Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Accepted Answer

Solution: (i) 140 We use the prime factorisation method: $$140 = 2 	imes 70$$ $$= 2 	imes 2 	imes 35$$ $$= 2 	imes 2 	imes 5 	imes 7$$ Final Answer: The prime factorisation of 140 is $2^2 	imes 5 	imes 7$. (ii) 156 We use the prime factorisation method: $$156 = 2 	imes 78$$ $$= 2 	imes 2 \...

Question 2

Q2: Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 33...

Accepted Answer

Solution: (i) 26 and 91 Given: Integers are 26 and 91. To Find: HCF and LCM, and verify the relationship. Solution: First, find the prime factorisation of each number: $$26 = 2 	imes 13$$ $$91 = 7 	imes 13$$ HCF is the product of the smallest power of each common prime factor. $$	ext{HCF}(26, 91)...

Question 3

Q3: Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

Accepted Answer

Solution: (i) 12, 15 and 21 Given: Integers are 12, 15, and 21. To Find: HCF and LCM. Solution: Prime factorisation: $$12 = 2 	imes 6 = 2^2 	imes 3$$ $$15 = 3 	imes 5$$ $$21 = 3 	imes 7$$ HCF is the product of the smallest power of each common prime factor. The only common prime factor is 3. $$\...

Question 4

Q4: Given that $\operatorname{HCF}(306,657)=9$, find $\operatorname{LCM}(306,657)$.

Accepted Answer

Given: The two numbers are 306 and 657. $$	ext{HCF}(306, 657) = 9$$ To Find: $$	ext{LCM}(306, 657)$$ Formula: For any two positive integers a and b, we have: $$	ext{HCF}(a, b) 	imes 	ext{LCM}(a, b) = a 	imes b$$ Solution: Using the formula, we can write: $$	ext{LCM}(306, 657) = \frac{306 	im...

Question 5

Q5: Check whether $6^{n}$ can end with the digit 0 for any natural number $n$.

Accepted Answer

To Check: Whether $6^n$ can end with the digit 0 for any natural number $n$. Reasoning: A number ends with the digit 0 if it is divisible by 10. This means its prime factorisation must include both 2 and 5. Solution: Let us find the prime factors of $6^n$. $$6^n = (2 	imes 3)^n = 2^n 	imes 3^n$$ T...

Question 6

Q6: Explain why $7 	imes 11 	imes 13+13$ and $7 	imes 6 	imes 5 	imes 4 	imes 3 	imes 2 	imes 1+5$ are composite numbers.

Accepted Answer

To Explain: Why the given expressions represent composite numbers. Definition: A composite number is a positive integer that has at least one divisor other than 1 and itself. Explanation: First number: $7 	imes 11 	imes 13 + 13$ We can take 13 as a common factor from the expression: $$7 	imes 11...

Question 7

Q7: There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Supp...

Accepted Answer

Given: Time taken by Sonia for one round = 18 minutes. Time taken by Ravi for one round = 12 minutes. They start at the same point, at the same time, and go in the same direction. To Find: The time after which they will meet again at the starting point. Solution: To find the time when they will meet...

Question 8

Q1: Prove that $\sqrt{5}$ is irrational.

Accepted Answer

To Prove: $\sqrt{5}$ is an irrational number. Proof: We will use the method of proof by contradiction. Let us assume, to the contrary, that $\sqrt{5}$ is a rational number. If it is rational, it can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers, $b 
eq 0$, and $a$ and $b$ a...

Question 9

Q2: Prove that $3+2 \sqrt{5}$ is irrational.

Accepted Answer

To Prove: $3 + 2\sqrt{5}$ is an irrational number. Proof: We will use the method of proof by contradiction. Let us assume, to the contrary, that $3 + 2\sqrt{5}$ is a rational number. If it is rational, it can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b 
eq 0$. $$3...

Question 10

Q3: Prove that the following are irrationals : (i) $\frac{1}{\sqrt{2}}$ (ii) $7 \sqrt{5}$ (iii) $6+\sqrt{2}$

Accepted Answer

Solution: (i) Prove that $\frac{1}{\sqrt{2}}$ is irrational. Proof: Let us assume, to the contrary, that $\frac{1}{\sqrt{2}}$ is rational. Then, we can find coprime integers $a$ and $b$ ($b 
eq 0$) such that: $$\frac{1}{\sqrt{2}} = \frac{a}{b}$$ Rearranging the equation, we get: $$\sqrt{2} = \frac{...

NCERT Solutions